where \(\sigma (\Gamma )\)  denotes the spectrum of \(\Gamma \) . By the choice of radii \(\delta _j / 2\) 's that define the circles \(\mathcal {B}_j\) 's, any point \(z \in \mathcal {B}_j\)  is not an eigenvalue of \(\Gamma \) , which by definition implies \(z\)  is in the resolvent set of \(\Gamma \) . Hence the first inequality of (REF ) follows from standard results on the norm of a resolvent (e.g. [1]}). The equality \(\mathrm {dist}(z, \sigma (\Gamma )) = \delta _j / 2\)  in (REF ) follows immediately by again the definition of \(\mathcal {B}_j\) .
